Abstract
In this work, ZZ transformation is combined with the Adomian decomposition method to solve the dynamical system of fractional order. The derivative of fractional order is represented in the Atangana–Baleanu derivative. The numerical examples are combined for their approximate-analytical solution. It is explored using graphs that indicate that the actual and approximation results are close to each other, demonstrating the method’s usefulness. Fractional-order solutions are the most in line with the dynamics of the targeted problems, and they provide an endless number of options for an optimal mathematical model solution for a particular physical phenomenon. This analytical approach produces a series form solution that is quickly convergent to exact solutions. The acquired results suggest that the novel analytical solution technique is simple to use and very successful at assessing complicated problems that arise in related fields of research and technology.
1. Introduction
Because of their extensive applications in many science and engineering disciplines, fractional differential equations have sparked much attention in recent years. Critical phenomena well characterize differential equations of fractional order in electromagnetics, finance, viscoelasticity, acoustics, material science, and electrochemistry. Barkai et al. [1], Mainardi [2], Tadjeran and Meerschaert [3], Meerschaert et al. [4], and Magin et al. [5] just released a review article on fractional signals and systems, including control theory applications. The edited volume of Machado contains several applications of fractional calculus, such as image processing [6]. The importance and necessity of fractional calculus can be seen in several applications in transdisciplinary disciplines. Miller and Ross [7], Oldham and Spanier [8], Podlubny [9], Kilbas et al. [10], Samko et al. [11], Caponetto [12], and Diethelm [13] have all authored essential studies on the fractional derivative and fractional differential equations. A review study on the recent history of fractional calculus was written by Machado et al. [14]. An article on recent developments in the theory of abstract differential equations with fractional derivatives was published by Hernandez et al. [15]. These publications provide a systematic explanation of fractional calculus, including the existence and uniqueness of solutions and various analytical methods for solving fractional differential equations, such as Green’s function method, power series approach, Mellin transform method, and others. No method in the literature produces a precise solution for nonlinear fractional differential equations (16) and (17). Using linearization or perturbation approaches, only approximate answers can be obtained. All of these push us to develop a numerical approach for fractional differential equations that is both efficient and accurate [18–21]. Chaos theory, heat transfer, variational issues, and other fields have used the Atangana–Baleanu fractional differential extensively. Recently, a fractional-differential mask based on a fractional Gaussian kernel with Atangana–Baleanu fractional differential has been published in the literature for the detection of blood vessels in retinal pictures, with the suggested method’s efficacy compared to other well-known approaches. Furthermore, it discusses the underlying differences between power-law, exponential-law, and Mittag–Leffler kernels, as well as their potential applications in diverse domains.
This paper establishes a connection between the Aboodh transformation (AT), ZZ transformation (ZZT), and Laplace transformation (LT), with several applications mentioned in [22–24]. The ZZT was then employed to define fractional Atangana–Baleanu Caputo operators and characterize Riemann–Liouville senses using theorems. Later, we solved several test problems stated in the Atangana–Baleanu sense using this ZZ transform. The current author’s contributions to this study are (i) applying the ZZ transform to solve fractional differential equations expressed in the Atangana–Baleanu derivative and (ii) establishing the connection between the Laplace, Aboodh, and ZZ transformations. A few well-known transforms that the ZZ transform generalizes can be related to other well-known transforms. Divide the ZZ transform by the adjusted variable to get the natural transform. Relationships with other integral transformations are also included in this work in terms of theorems. This transformation has the advantage of converging to the Sumudu transformation, which is advantageous when solving fractional differential equations with variable coefficients, such as [25–27].
Adomian (1980) established the Adomian decomposition technique (ADM), an efficient method for finding explicit and numerical solutions to a larger and more general class of differential systems representing real-world issues [28–30]. This strategy effectively addresses initial and boundary value problems, linear and nonlinear, ordinary and partial differential equations, and stochastic systems. Furthermore, this approach does not require any linearization or perturbation. ADM has been used extensively in the last two decades since it yields approximate analytical solutions for nonlinear problems, and there has been much interest in utilizing it to solve fractional differential equations (31)–(33).
The ZZ decomposition method has the following advantages with respect to Adomian decomposition method:(i)ZZ decomposition method required small calculations as compared to the Adomian decomposition method(ii)The fractional derivative is simplified by using the ZZ transformation first and then applying decomposition method while it is not the case if we use the Adomian decomposition method directly(iii)The initial conditions/boundary conditions are used directly in the ZZ decomposition method, and it mostly avoids the extra calculations of Adomian polynomials
The novelty of the present work is to deal with the analytical solutions of important fractional-order some dynamical systems. The fractional-order of some dynamical systems have many applications in physical sciences, and therefore, different graphs are presented to show various dynamics of fractional parabolic equations. The solutions are obtained in rather simpler way as compared to other techniques. The current study has been structured as follows: in Section 2, some basic notions of basic definitions of ZZ transformation are described. In Section 3, we give an analysis of the suggested technique. In Section 4, we provide current solutions which suggested equations explaining how to implemented the suggested technique. Finally, the conclusion is provided.
2. Preliminaries
Definition 1. The function set of the Aboodh transform (AT) is defined asand is given as [22, 23]
Theorem 1. Now, we consider and as the Aboodh and Laplace transforms of ; then [24, 25],Zafar [26] was the first to develop the ZZ transform. It is mixture of the Laplace and Aboodh integral transforms. The ZZ transform is expressed in the following.
Definition 2. (ZZ transform). Suppose that is a function, then the ZZ transformation of is defined as [26]The ZZ transformation is linear, just as the Aboodh and Laplace transforms. The MLF is a function that is given as an extension of the exponential term.
Definition 3. The Atangana–Baleanu Caputo derivative of a function ; then, for , it is defined as [27]
Definition 4. Let the Riemann–Liouville Atangana–Baleanu derivative ; then, for , it is given as [27]with the condition , is a function, and .
Theorem 2. The Laplace transform of Riemann–Liouville Atangana–Baleanu derivative and Atangana–Baleanu Caputo are, respectively, defined as [27]andThe theorems that follow are based on the idea that and .
Theorem 3. The Aboodh transformation of Atangana–Baleanu Riemann–Liouville derivative is defined as [25]
Proof 1. Using Theorem 1 and equation (3), we arrive to the required solution. The relationship among the transforms of ZZ and Aboodh is given in the following theorem.
Theorem 4. The Aboodh transform of Atangana–Baleanu Caputo derivative is defined as [25]
Proof 2. Using Theorem 1 and equation (2), we can discover the desired solution.
Theorem 5. If and are the Aboodh and ZZ transforms of , then we obtain the following [25]:
Proof 3. (The transform definitions). We getPut in (13); we getThe right-hand side of (14) may be expressed aswhere expresses the Laplace transformation of . Using Theorem 1, (15) can be defined aswhere defines the Aboodh transform of .
Theorem 6. transformation of is defined as
Proof 4. The Aboodh transformation of , isApplying (17), we achieve
Theorem 7. Let and ; then, the transformation of is defined as [25]
Proof 5. We know that Aboodh transform of is defined asSo,Applying Theorem 9, we achieve
Theorem 8. If and are the Aboodh and ZZ transforms of , then the Atangana–Baleanu Caputo ZZ transformation derivative is defined as [25]
Proof 6. Applying equations (1) and (5), we getSo, the Atangana–Baleanu Caputo of transformation is defined as
Theorem 9. Let us suppose that and are the Aboodh and ZZ transforms of . Then, the Atangana–Baleanu Riemann–Liouville ZZ transform derivative is defined as [25]
Proof 7. Applying equations (1) and (4), we getFrom (16), the ZZ transform of Riemann–Liouville Atangana–Baleanu is defined as
3. Idea of MDM
Consider the fractional order partial differential equation by MDM:with the initial conditionwhere is the Atangana–Baleanu fractional derivative of order ; is linear and nonlinear terms, respectively. On both sides, we use ZZ transformation of (30), to achieve
By the differentiation property of ZZ transformation, we get
(33) implies that
Applying the ZZ inverse transformation of (34), we get
MDM determines the infinite sequence’s result of :
The nonlinear functions can be found with the help of Adomian polynomials which is expressed as
The Adomian polynomials can show all types of nonlinearity as
Putting (36) and (38) into (35), it gives
The following terms are described:
The general form for is determined as
4. Numerical Examples
Example 1. Here, we take the following FPDE:with initial sourceThe exact result at is (1) and (2) .
Applying ZZT (42), we getWe getApplying the inverse ZZT to (45), we getDecomposition results for and can be expressed asFurthermore,The component comparison in (48) provides the following recursive MDM algorithm:For ,For ,For ,Similar to , MDM can be used to determine the remaining terms of and . In general, MDM’s solution is as follows:Set in (42); we getThe exact results are at .We analyze the solution figures of the problem, which have been investigated by applying the ZZ decomposition method in the sense of the Atangana–Baleanu operator. Figure 1 represents the three-dimensional solution-figures for variables of example 1 at fractional order and 0.8, respectively; Figure 2 represents different fractional order of and 0.4; and Figure 3 represents that at . In Figure 4, different fractional order with respect to and . It is observed that the ZZ decomposition method solution-figures are identical and in close contact with each other. In the same way, Figures 5–8 show different fractional order graphs of at of Example 1.
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Example 2. Here, we take the following FPDE:with initial sourcesThe exact solution at isUsing ZZT equation (33), it can be written asAfter simplification, we obtainTaking inverse ZZT of (60), we obtainAssume decomposition solutions for variables , , and , it can be written asRemember that , , , , and are the Adomian polynomials and the nonlinear terms were characterized, which can be further simplified asUsing (38), the nonlinearity in the given problem can be expressed asThe component comparison provides the following recursive MDM algorithm:For ,For ,For ,In same manner, the remaining terms of , and for can be calculated easily by using MDM. The general solution of MDM is given bySetting in (69), we getwhich is the MDM solution in closed form of equation (34). When ,We analyze the solution-figures of the problem, which have been investigated by applying the ZZ decomposition method in the sense of the Atangana–Baleanu operator. Figure 9 represents the two-dimensional solution-figures for variables of example 2 and second graph of different fractional order . Figure 10 represents the two-dimensional solution-figures for variables of example 2 and second graph of different fractional-order . Figure 11 represents the two-dimensional solution-figures for variables of example 2 and second graph of different fractional-order . It is observed that the ZZ decomposition method solution-figures are identical and in close contact with each other.
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5. Conclusion
In this paper, some important system of fractional partial differential equations is considered for its analytical solution using the ZZ decomposition method. It has been demonstrated from the figures that the present techniques have the greater tendency to analyze the results of the given models. The problems results at different time fractional are investigated which cover the various aspects of the proposed models and proposed method. The results at different fractional orders are suggested and shown a very closed convergence phenomena of the fractional results towards integer order solutions. The graph has shown a very consistent relation between the integer and fractional orders results. It is noted that the effective and straight-forward solution of the ZZ decomposition method implies its applicability to solve other fractional partial differential equations.
Data Availability
The numerical data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work under Grant Code number: 22UQU4310396DSR18.